6) Nature
Plants and Beasts and Fungus, oh my! Add a little healing and a little control to split the difference.
Now you're not gonna launch a new setting with one combat role for each of these power sources, obviously. That's 30 classes out of the gate before we even get to theming, which is the next most important step. But you should probably try to have at least one of each in the setting to make sure you've got somewhere for pretty much everyone to play. At least one Nature, at least one Arcane, etc. That said, Psychic is always the easiest to cut because people have the least amount of investment into it, lately.
But I did mention Theming... and this one is -so- important, you guys. It is the kind of thing that will define the game table in your settings. The types of characters people like to make.
1) Pretty/Aesthetic
Sounds like a cop-out but it really isn't. You're going to see the faerie princess pop up at your table a fair amount of times. It doesn't specifically have to be a faerie or a princess, but there's going to be people who want to play the 'pretty' class. This is why Warlocks have a Fey Patron option that gets glamours and stuff instead of Fey Patrons that give you decay and rot powers. Bard is probably the quintessential 'pretty' class. You know who else can be pretty? Monks. Robes and peace and gentleness that turns into a fist of iron in the middle of the tea party ceremony.
2) Heroic/Normal
Heroic coves a -lot- of what pretty does, but it also covers your knights in shining armor and some less armored types like barbarians and the grandstanding gladiators or warlords. It also applies to swashbucklers of every stripe, most of the ranger concepts that don't sit in the corner, brooding with a pipe. Your heroic players are going to want classes that stand out with big main character energy. Generally speaking, this won't be a spellcaster, but instead someone who hits things.
3) Dark/Strange
Vampire the Masquerade has Nosferatu as a clan because sometimes people want to play the monstrous even in a game about monsters. Heroic monsters, but monstrous nonetheless. This is why Warlocks exist for the most part. But you can also do dark Rangers, Barbarians, and Rogues pretty easily. Monstrous ones, too. Most settings and games tend to make the dark or strange into a species choice, almost exclusively, and maybe toss in some small asides for magic items or optional features that add some dark/strange.
Now if you were to do a PHD (Pretty/Heroic/Dark) for each of the 5 roles you'd wind up with 15 classes. Much more manageable, but still more than WotC's core class list of 12. 13 if you include Artificer, 14 if you include Blood Hunter, and bang on 15 if you include the Illrigger.
So let's look at a potential example list:
1) Tanks
a) Champion. Martial Heroic type.
b) Swordmage. Arcane Pretty type.
c) Blackguard. Occult Dark type.
2) Skirmishers
a) Keeper. Occult Heroic type.
b) Berserk. Nature Pretty type.
c) Inquisitor. Divine Dark type.
3) Blasters
a) Alchemist. Martial Heroic type.
b) Primalist. Nature Pretty type.
c) Esper. Psychic Dark type.
4) Support
a) Captain. Martial Heroic type.
b) Priest. Divine Pretty type.
c) Spiritualist. Occult Dark type.
5) Snipers
a) Warcaster. Arcane Heroic type
b) Minstrel. Arcane Pretty type.
c) Witch. Nature Dark type.
Yes, some of the above classes are classes I've put out for A5e and 5e compatible games. Don't read too much into that.
With this setup you wind up with 3 Arcane, 3 Martial, 3 Nature, 3 Occult, 2 Divine, and 1 Psychic. As a bonus, I unintentionally excluded any Arcane Full Casters which, hey, really limits the existence of some of the more problematic spells. Win/Win, there, for setting yourself up for an easier time building a more balanced system tied to your setting.
As a reminder, in this thought experiment we're -not- doing the core classes, since you're basically piloting a new functionality built around the 'full' adventuring day that D&D 5e is built off of.
Thoughts? Comments?